### What is an Inverse Problem?

#### An inverse problem is determining unknown *causes* based on observation of their *effects*.

If the following two questions are both answered ‘yes’, then you have an inverse problem:

• Am I using a mathematical model?

• Am I using data?

Inverse problems include both parameter estimation and function estimation. Inverse problems “bridge” experimental studies and data reduction.

Inverse problems have a wide range of applications, such as medical imaging (MRI), oil drilling, echolocation (SONAR) and thermal properties measurement. A common characteristic is that we attempt to infer causes (input) from measured effects (output). Because the measured data is often noisy or indistinct, the optimal or ``best’’ prediction to the inverse problem may be difficult to obtain owing to the ill-posed nature of the problem.

An example of an inverse problem is estimating the thermal diffusivity of a solid from transient temperature measurements, using the heat diffusion PDE. The corresponding forward problem is to compute the solid temperature using a known thermal diffusivity. Commercial finite-element programs are typically forward-problem solvers, and require the parameter(s) to be known.

Estimating heat flux at a solid surface from measured temperatures within the solid is an example of function estimation.

### Background of the Inverse Problems Conference

In engineering science, a physical process is typically analyzed using a mathematical model in which the actual system is represented by a set of equations containing parameters. The classical direct problem is to find the output of the system given the input and the system parameters. Inverse problems, on the other hand, involve determining the unknown causes of known consequences. There are two main types: 1) the input estimation problem where the system parameters and output are known and for which the missing part of the input (boundary or initial conditions) are to be determined, and 2) the identification or parameter estimation problem, where the parameters are found given the input and output. Note that the given output can either be a measured response of the system (inverse problem) or a desired response (inverse design problem). All inverse problems are classified as ill-posed in that their solutions do not necessarily satisfy conditions of existence, uniqueness, and stability.

Inverse problems are interdisciplinary and have applications in a wide range of fields in engineering. Many of the mathematical methods employed in inverse problems are similar to those used in control and design, but what makes the inverse problem much more challenging than those applications is the ill-posedness of the problem. The problems are ill-posed primarily because the solutions are continuously dependent on the data, meaning that a small change in the measured data can cause a large change in the solution of the inverse problem. Since inverse problem solutions necessary rely on measured data, the ill-posedness makes the solutions more difficult than control or design problems.

Finally, a key application area for inverse problems is the optimal design of experiments. Experimental data are often acquired in order to estimate one or more parameters of interest, such as materials physical properties, thermomechanical boundary conditions, geophysical properties from satellite data, and metrics used to quantify a biophysical condition to help medical diagnostics. By understanding how parameters of interest can be estimated via an inverse problem, one can optimally design his or her experiment to best estimate those parameters of interest, in the sense that the experimental noise will minimally pollute the estimation. Given the high cost of large-scaled experiments, many disciplines can benefit from the development of inverse problem techniques.

### History

In each year without an international conference, informal two-day seminars (Inverse Problem Symposia) have been held at various universities in the US. These sites and organizers are listed along with the international conferences in Table 1. Michigan State University has been prominent among these, and continues to be.

Previous international meetings were held in 1993 at Palm Coast, FL, in 1996 at Le Croisic, France, in 1999 at Port Ludlow, WA, in 2002 at Angra dos Reis, Brazil, and in 2005 at Claire College, Cambridge, England. These international conferences grew out of several informal meetings held in the years prior to the Palm Coast conference, which focused mainly on inverse problems in heat transfer. Nicholas Zabaras broadened the scope of the Conference in the first international conference in 1993, especially into the field of solid mechanics. This conference was successful in attracting participation from the mechanics group, and drew in total about 40 participants. Typical conference numbers range from 50-100 attendees.

Click here to see list of past conferences.

### Committees

**Steering Committee**

**IPS Organizing Committee and Contracts**

- Dr. Jay I. Frankel, Chairperson, R.G. Myers Endowment Professor and Department Head of Department of Mechanical and Aerospace Engineering at NMSU < jfrankel@nmsu.edu >
- Ms. Victoria Trujillo, Conference Coordinator <vt1@nmsu.edu >